The expression you gave is close (you missed the factor $\frac{1}{\sqrt{2\pi}}$) to the correct asymptotic behavior of the survival function of standard normal distribution:  
\begin{align*}
1 - \Phi(x) \sim \frac{1}{x}\varphi(x), \tag{1}
\end{align*}
where $\Phi$ and $\varphi$ are CDF and PDF of standard normal random variable respectively.  

$(1)$ is a corollary of the inequality (for fixed $x > 0$)
\begin{align*}
(x^{-1} - x^{-3})\varphi(x) < 1 - \Phi(x) < x^{-1}\varphi(x). \tag{2}
\end{align*}

To prove $(2)$, notice the obvious inequality: 
\begin{align*}
(1 - 3t^{-4})\varphi(t) < \varphi(t) < (1 + t^{-2})\varphi(t), \; t > x. 
\end{align*}
Integrating three expressions above from $x$ to $\infty$ yields $(2$).